Why Mathematical Programming is Useful

to advertise the relaunch of the Mathematical Programming Study Group;

to encourage you to use Mathematical Programming in your work.

What is Mathematical Programming?

First
a definition: Mathematical Programming (MP) is the use of mathematical
models, particularly optimizing models, to assist in taking decisions.

The
term 'Programming' antedates computers and means 'preparing a schedule
of activities'. It is still used, for instance, in oil refineries,
where the refinery programmers prepare detailed schedules of how the
various process units will be operated and the products blended.
Mathematical Programming is, therefore, the use of mathematics to
assist in these activities.

Linear Programming

One
special case of Mathematical Programming which has been enormously
successful is Linear Programming (LP). In an LP model all the
relationships are linear, hence the name. LP has been so successful for
two reasons:

there are robust 'black box' solvers which find the best solution to LP problems automatically;

many real-world phenomena can be approximated reasonably well by linear relationships.

However,
Mathematical Programming is not just LP and its extensions, such as
Integer Programming and Quadratic Programming. It also embraces
heuristics and algorithms designed to tackle specific problems. The key
characteristics of MP are shown in the diagram below.

MP Models

The mathematical modeller abstracts from the real world a model of the system under investigation. The model comprises:

a set of decision variables, whose values are to be determined;

the relationships between them, the constraints;

a means of comparing the quality of solutions which satisfy the constraints, the objective.

The
definitions of all these components will change repeatedly during the
building of the model. Although the process of MP involves finding
optimum solutions, noby is suggesting that the solution is optimum to
the real-world problem.

If the model is reasonably faithful, then its optimum solution should be a good
solution to the real-world problem. Whether it is or not, the process
of building the model and analysing the solutions is a very powerful
tool in analysing the real-world problem.

Building an MP Model

Optimization
exercises a model in a far more rigorous way than other techniques,
such as simulation. The optimum solution is, by definition, extreme.
When building an MP model one is reminded of the proverb:

Man proposes: God disposes

One
builds the model and then turns it over to the optimization algorithm
to find the best solution. If there is a fault in the model which can
be exploited, the optimization algorithm will find it. The solution
will be, at best, impracticable, at worst, nonsense. But through such
mistakes one acquires greater understanding of the problem and moves
towards solutions which are truly useful.

Another
aspect of optimization is its need for data. Some argue that this is a
weakness, that you can only use the technique when all the data are
available. But the requirement for data imposes a useful discipline. If
the data do not exist to support some representation of the problem,
there is no point using it. No conclusions will ever be able to be
drawn. Instead, the representation of the problem must be changed to
reflect the data which are available (or which can be estimated).

The
process of building an optimization model is therefore necessarily
iterative. A first draft of the model will be sketched out and test
data sought. Most probably some of the data will prove to be
unobtainable. The model has to be altered before it can be run. The
solutions are nonsense. Some constraint has been omitted. It is added.
The solution is now plausible given the test data and the highly
simplified representation.

More
detail is added to the model. Further data are sought. So the process
goes on. As the model gets better, so the client's scepticism gives way
to enthusiasm. The model starts to propose new ways of doing things. It
is really beginning to add value.

Ultimately
the time comes when the model moves from being an experimental tool to
a decision support aid. This is when the user interface becomes
critical. Fortunately, MP software has moved forward a lot in the past
few years and MP models can now be embedded straightforwardly in larger
systems, taking their data from databases and spreadsheets and
returning their results there.

The MP Study Group

What does this all mean for the MP Study Group? Two things:

meetings will be primarily devoted to presentations about practical applications;

everyone who is interested in practical Operational Research is likely to find them stimulating.

The
first meeting of the relaunched MP Study Group will be at 18.30 on
Wednesday 5th October 1994 at Friends House, Euston Road, London
(opposite Euston Station). Two talks will be presented. These
illustrate different approaches to tackling practical problems which
arose in agriculture but which are similar to many throughout the
transport and distribution sector. Both involve working out patterns of
collection for products where regular visits must be made to particular
locations, in one case for eggs, and in the other for milk.

Finally,
if you have worked on any models which you think would be of wider
interest and which you would like to present at a Study Group meeting,
please get in touch.